Recall that state i is said to be positive recurrent if mi, i<∞, where mi, i is the expected num

Step 1: Recall that state $i$ is positive recurrent if $m_{i, i} < infty$, where $m_{i, i}$ is t

Question

Recall that state $i$ is said to be positive recurrent if $m_{i, i}<\infty$, where $m_{i, i}$ is the expected number of transitions until the Markov chain, starting in state $i$, makes a transition back into that state. Because $\pi_{i}$, the long run proportion of time the Markov chain, starting in state $i$, spends in state $i$, satisfies $$ \pi_{i}=\frac{1}{m_{i, i}} $$ it follows that state $i$ is positive recurrent if and only if $\pi_{i}>0$. Suppose that state $i$ is positive recurrent and that state $i$ communicates with state $j .$ Show that state $j$ is also positive recurrent by arguing that there is an integer $n$ such that $$ \pi_{j} \geqslant \pi_{i} P_{i, j}^{n}>0 $$

          Recall that state $i$ is said to be positive recurrent if $m_{i, i}<\infty$, where $m_{i, i}$ is the expected number of transitions until the Markov chain, starting in state $i$, makes a transition back into that state. Because $\pi_{i}$, the long run proportion of time the Markov chain, starting in state $i$, spends in state $i$, satisfies
$$
\pi_{i}=\frac{1}{m_{i, i}}
$$
it follows that state $i$ is positive recurrent if and only if $\pi_{i}>0$. Suppose that state $i$ is positive recurrent and that state $i$ communicates with state $j .$ Show that state $j$ is also positive recurrent by arguing that there is an integer $n$ such that
$$
\pi_{j} \geqslant \pi_{i} P_{i, j}^{n}>0
$$

Added by James H.

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